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Theorem imaiun1 39981
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3247 . . . 4 (∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2 vex 3496 . . . . . 6 𝑦 ∈ V
32elima3 5929 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
43rexbii 3245 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5 eliun 4914 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
65anbi2i 624 . . . . . 6 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
7 r19.42v 3348 . . . . . 6 (∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
86, 7bitr4i 280 . . . . 5 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
98exbii 1841 . . . 4 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
101, 4, 93bitr4ri 306 . . 3 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
112elima3 5929 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵))
12 eliun 4914 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
1310, 11, 123bitr4i 305 . 2 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ 𝑦 𝑥𝐴 (𝐵𝐶))
1413eqriv 2816 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1530  wex 1773  wcel 2107  wrex 3137  cop 4565   ciun 4910  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-iun 4912  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  trclimalb2  40056
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